I can’t think of any other factors besides sample patch and transposition. Both “Horn” and “French horn” provide the same list of available transposition variants in Setup mode.
As you know, “French horn” is a bit of an anachronism, and “horn” is the more commonly preferred modern term. Dorico probably just provides it as a naming option in setup for those who still prefer that term.
The name “French horn” is found only in English, first coming into use in the late 17th century. At that time, French makers were preeminent in the manufacture of hunting horns, and were credited with creating the now-familiar, circular “hoop” shape of the instrument. As a result, these instruments were often called, even in English, by their French names: trompe de chasse or cor de chasse (the clear modern distinction between trompes, trumpets, and cors, horns, did not exist at that time). German makers first devised crooks to make such horns playable in different keys—so musicians came to use “French” and “German” to distinguish the simple hunting horn from the newer horn with crooks, which in England was also called by the Italian name corno cromatico (chromatic horn). More recently, “French horn” is often used colloquially, though the adjective has normally been avoided when referring to the European orchestral horn, ever since the German horn began replacing the French-style instrument in British orchestras around 1930. The International Horn Society has recommended since 1971 that the instrument be simply called the horn.
There is also a more specific use of “French horn” to describe a particular horn type, differentiated from the German horn and Vienna horn. In this sense, “French horn” refers to a narrow-bore instrument (10.8–11.0 mm [0.43–0.43 in]) with three Périnet (piston) valves. It retains the narrow bell-throat and mouthpipe crooks of the orchestral hand horn of the late 18th century, and most often has an “ascending” third valve. This is a whole-tone valve arranged so that with the valve in the “up” position the valve loop is engaged, but when the valve is pressed the loop is cut out, raising the pitch by a whole tone.